17TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2016 ISGG
4–8 AUGUST, 2016, BEIJING, CHINA
Paper #008
PERIODIC FRACTAL PATTERNS
Douglas DUNHAM1, and John SHIER2
1University of Minnesota, USA 2Apple Valley Minnesota, USA
ABSTRACT: We present an algorithm that can create patterns that are locally fractal in nature, but
repeat in two independent directions in the Euclidean plane — in other word “wallpaper patterns”.
The goal of the algorithm is to randomly place progressively smaller copies of a basic sub-pattern or
motif within a fundamental region for one of the 17 wallpaper groups. This is done in such a way
as to completely fill the region in the limit of infinitely many motifs. This produces a fractal pattern
of motifs within that region. Then the fundamental region is replicated by the defining relations of
the wallpaper group to produce a repeating pattern. The result is a pattern that is locally fractal, but
repeats globally — a mixture of both randomness and regularity. We show several such patterns.
Keywords: Fractals, wallpaper groups, algorithm
1. INTRODUCTION
We have previously described how to fill a pla-
nar region with a series of progressively smaller
randomly-placed sub-patterns or motifs produc-
ing pleasing patterns [2, 5]. In this paper we ex-
plain how the basic algorithm can be modified
to fill fundamental regions of wallpaper groups,
and thus create wallpaper patterns. Figure 1
shows a random circle pattern with symmetry
group p1. Thus to create our patterns, we first
fill a fundamental region for one of the 17 2-
dimensional crystallographic groups (or wallpa-
per groups) with randomly placed, progressively
smaller copies of a motif, such as the circles in
Figure 1. Then we apply transformations of the
wallpaper group to the fundamental region in or-
der to tile the plane. This produces a locally ran-
dom, but globally symmetric pattern.
In the next section we describe how the ba-
sic algorithm works. Then we recall a few facts
about wallpaper groups, and indicate how the
alorithm can be modified to produce filled funda-
mental regions for those groups. Next we exhibit
some patterns. Finally, we draw conclusions and
summarize the results.
Figure 1: A circle pattern with p1 symmetry.
2. THE ALGORITHM
The algorithm works by placing a sequence of
progressively smaller sub-patterns or motifs mi
within a region R so that a motif does not overlap
any previously placed motif. Random locations
are tried until a non-overlapping one is found.
This process continues indefinitely if the motifs
adhere to an “area rule” which is described fol-
lowing the algorithm:
For each i = 0,1,2, . . .
Repeat:
Randomly choose
a point within R to
place the i-th mo-
tif mi.
Until (mi doesn’t intersect
any of m0,m1, ...,mi−1)
Add mi to the list of success-
ful placements
Until some stopping condition is met,
such as a maximum value of i or a min-
imum value of Ai.
It has been found experimentally by the second
author that the algorithm does not halt for a
wide range of choices of shapes of R and the
motifs provided that the motifs obey an inverse
power area rule: if A is the area of R, then for
i = 0,1,2, . . . the area of mi, Ai, can be taken to
be:
Ai =A
ζ (c,N)(N + i)c(1)
where c > 1 and N > 1 are parameters, and
ζ (c,N) is the Hurwitz zeta function: ζ (s,q) =
∑∞k=0
1(q+k)s . Thus limn→∞ ∑n
i=0 Ai = A, that is,
the process is space-filling if the algorithm con-
tinues indefinitely. In the limit, the fractal di-
mension D of the placed motifs can be com-
puted to be D = 2/c [5]. Examples of the al-
gorithm written in C code can be found at the
second author’s web site [6]. It is conjectured
by the authors that the algorithm does not halt
for non-pathological shapes of R and mi, and
“reasonable” choices of c and N (depending on
the shapes of R and the mis). In fact this has
been proved for 1 < c < 1.0965... and N ≥ 1 by
Christopher Ennis when R is a circle and the mo-
tifs are also circles [3].
3. WALLPAPER PATTERNS
It has been known for more than a century that
there 17 different kinds of patterns in the Eu-
clidean plane that repeat in two independent di-
rections. Such patterns are called wallpaper pat-
terns and their symmetry groups are called plane
crystallographic groups or wallpaper groups. In
1952 the International Union of Crystallography
(IUC) established a notation for these groups. In
1978 Schattschneider wrote a paper clarifying
the notation and giving an algorithm for identi-
fying the symmetry group of a wallpaper pattern
[4]. Later, Conway popularized the more general
orbifold notation [1]. So, as mentioned above,
we create fractal wallpaper patterns by first fill-
ing a fundamental region R for a wallpaper group
with motifs, then extend the pattern using trans-
formations of the wallpaper group. Figures 1
(above) and 2 show patterns that have p1 (or o in
orbifold notation) symmetry, the simplest kind
of wallpaper symmetry, with only translations in
two independent directions. In Figure 2 notice
that the peppers on the left edge “wrap around”
and are continued on the right edge; similarly
peppers on the top edge “wrap around” to the
bottom.
Figure 2: A pattern of peppers with p1 symme-
try.
We consider two issues that arise with motifs
being placed in a fundamental region for a wall-
paper group: (1) what to do if a motif crosses a
mirror line that is the boundary of the region, and
(2) what to do if the motif overlaps a center of ro-
tation. One solution is to just let that happen, as
2
is shown in Figure 3 for circles overlapping mir-
ror lines. Another solution is to simply reject all
Figure 3: Overlapping circles on mirror bound-
aries.
motifs that cross a mirror boundary or overlap a
rotation point. Figure 4 shows a pattern of hearts
with p2mm symmetry that avoids the reflection
lines.
Figure 4: A pattern of hearts that avoids mirror
boundaries.
If a fundamental region has a mirror boundary
and the overlapping motif itself has mirror sym-
metry, then a more satisfactory solution is that
we move the motif (perpendicularly) onto the
boundary so that the mirror of the motif aligns
with the boundary mirror. Also, the area rule cal-
culation needs to be adjusted each time this hap-
pens since only half of the motif is placed within
the fundamental region. Figure 5 shows a frac-
tal pattern of flowers with p6mm symmetry with
flowers centered on the mirror lines.
Figure 5: A pattern of flowers with some on mir-
ror boundaries.
Similarly if the fundamental region has a ro-
tation point and the overlapping motif has that
kind of rotational symmetry, we move the mo-
tif to be centered on the rotation point. Figure 6
shows a p4 pattern of circles with magenta cir-
cles on one kind of 4-fold rotation point (those
4-fold points are at the corners, centers of the
edges, and center of the figure). Again the area
Figure 6: A p4 pattern of circles with some cen-
tered on 4-fold rotation points.
rule calculation needs to be adjusted if a motif
3
overlaps one of the 4-fold points or the 2-fold
point. Of course circles make excellent motifs
for testing these solutions since they have mirror
symmetry across any axis through the center and
n-fold rotational symmetry for any n.
4. SAMPLE PATTERNS
In this section, we present a few more patterns.
First, Figure 7 shows a pattern of black and
white triangles on a blue background with p4mm
symmetry. Figure 8 shows another triangle pat-
Figure 7: A triangle pattern with p4mm symme-
try.
tern, but with p6mm symmetry. Finally, Figure 9
Figure 8: A triangle pattern with p6mm symme-
try.
shows a pattern of arrows with p6mm symmetry.
Figure 9: An arrow pattern with p6mm symme-
try.
5. CONCLUSIONS
We have presented a geometric algorithm that
can be used to fill the fundamental region of one
of the 17 wallpaper groups with a sequence of
progressively smaller motifs. In turn that filled
fundamental region can be replicated about the
Euclidean plane to produce a repeating pattern
that is locally fractal.
In the future we hope to extend this concept to
hyperbolic patterns and patterns with color sym-
metry.
REFERENCES
[1] J. Conway, H. Burgiel, C. Goodman-
Strauss. The Symmetries of Things,
A.K. Peters, Ltd., Wellesley, MA, 2008.
ISBN 1-56881-220-5
[2] D. Dunham, J. Shier. The Art of Random
Fractals Bridges Conference Proceedings,
Seoul, Korea, 79–86, August, 2014. ISBN
978-1-938664-11-3
[3] C. Ennis. (Always) Room for One More
MAA Math Horizons, February, 2016.
[4] D. Schattschneider. The Plane Symme-
try Groups: Their Recognition and Nota-
4
tion American Mathematical Monthly, 85,
6, 439–450, July, 1978.
[5] J. Shier, P. Bourke. An Algorithm for Ran-
dom Fractal Filling of Space Computer
Graphics Forum, 32, 8, 89–97, December,
2013.
[6] J. Shier. Web site: http://www.john-
art.com/stat geom linkpage.html Last ac-
cessed 30 May, 2016.
ABOUT THE AUTHORS
1. Douglas Dunham is a Professor in the De-
partment of Computer Science at the Uni-
versity of Minnesota Duluth. His research
interests include repeating patterns, hyper-
bolic geometry, fractal patterns, and the
art of M.C. Escher. He can be reached
at his email address: [email protected]
or via his postal address: Department
of Computer Science, University of Min-
nesota, 1114 Kirby Drive, Duluth, Min-
nesota, 55812-3036, USA.
2. John Shier is a retired physicist. His re-
search interests include fractal patterns, al-
gorithms, and statistical geometry. He
can be reached at his email address:
[email protected] or via his postal
address: 6935 133rd Court, Apple Valley,
Minnesota, 55124, U.S.A.
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